Judgments and Propositions
Here the name ∧I stands for “conjunction introduction”, since the conjunction is introduced in the conclusion.
This rule allows us to conclude that A ∧ B true if we already know that
A true and B true. In this inference rule, A and B are schematic variables,
and ∧I is the name of the rule. The general form of an inference rule is
J1 . . . Jn
where the judgments J1 , . . . , Jn are called the premises, the judgment J is
called the conclusion. In general, we will use letters J to stand for judgments, while A, B, and C are reserved for propositions.
We take conjunction introduction as specifying the meaning of A ∧ B
completely. So what can be deduced if we know that A ∧ B is true? By the
above rule, to have a verification for A ∧ B means to have verifications for
A and B. Hence the following two rules are justified:
A ∧ B true
A ∧ B true
The name ∧EL stands for “left conjunction elimination”, since the conjunction in the premise has been eliminated in the conclusion. Similarly ∧ER
stands for “right conjunction elimination”.
We will see in Section 8 what precisely is required in order to guarantee
that the formation, introduction, and elimination rules for a connective fit
together correctly. For now, we will informally argue the correctness of the
elimination rules, as we did for the conjunction elimination rules.
As a second example we consider the proposition “truth” written as
>. Truth should always be true, which means its introduction rule has no
Consequently, we have no information if we know > true, so there is no
A conjunction of two propositions is characterized by one introduction
rule with two premises, and two corresponding elimination rules. We may
think of truth as a conjunction of zero propositions. By analogy it should
then have one introduction rule with zero premises, and zero corresponding elimination rules. This is precisely what we wrote out above.
L ECTURE N OTES
J ANUARY 12, 2010